Real-rooted atlas

A290596: Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

A290596 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A290596

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(6 + 6\,x\right)\,P_{n-1}'(x) + \left(9\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
31010121
480120241225
588017605284413213
6123203080012320154070157051
720944062832031416052360357010211207953
8418880014660800879648018326001666007140140129652561
996342400385369600269758720674396807663600437920128801841827024985
102504902400112720608009017648640263014752035865648025618320100464021528234125810060563
1172642169600363210848000326889763200108963254400173350632001485862560728364002081040339302901890601912621
12232454942720012785021849600127850218496004794383193600871706035200871706035205127682560183131520398112051040352133653167805313
13813592299520004881553797120005369709176832002237378823680004576456684800052302362112003589377792001538304768041801760071456007392042011381592958838821
14309165073817600020095729798144000241148757577728001105265138897920025119662247680003229670860416002533075184640012665375923204130013888088248160012172160103740494161216480478217195
151267576802652160008873037618565120001153494890413465600576747445206732800144186861301683200205981230430976001817481444979200103856082570240395104661952010130888768017467049601984892014177857412911014153736460073

b-files: A290596 ?

Technical details
Node id
n-33e7d067844832dd · revision 1 · JSON
Definition source
oeis_catalog:A290596
Fingerprint
f6-94f770baffeadbcb

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A290596 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          2,           1;
   3:         10,          10,          1;
   4:         80,         120,         24,          1;
   5:        880,        1760,        528,         44,         1;
   6:      12320,       30800,      12320,       1540,        70,        1;
   7:     209440,      628320,     314160,      52360,      3570,      102,       1;
   8:    4188800,    14660800,    8796480,    1832600,    166600,     7140,     140,     1;
   9:   96342400,   385369600,  269758720,   67439680,   7663600,   437920,   12880,   184,   1;
  10: 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) + 9*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (6 + 6*x)*D[P[n - 1, x], x] + 9*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (6 + 6*x)*deriv(v[i - 1], x) + 9*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A290596 triangle, offset 0, from P1
%e Example
Triangle begins:
  0:          1;
  1:          2,           1;
  2:         10,          10,          1;
  3:         80,         120,         24,          1;
  4:        880,        1760,        528,         44,         1;
  5:      12320,       30800,      12320,       1540,        70,        1;
  6:     209440,      628320,     314160,      52360,      3570,      102,       1;
  7:    4188800,    14660800,    8796480,    1832600,    166600,     7140,     140,     1;
  8:   96342400,   385369600,  269758720,   67439680,   7663600,   437920,   12880,   184,   1;
  9: 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) + 9*x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (6 + 6*x)*D[P[n - 1, x], x] + 9*x*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + (6 + 6*x)*deriv(v[i - 1], x) + 9*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.